Bose-Einstein condensation in complex networks

The evolution of many complex systems, including the world wide web, business and citation networks is encoded in the dynamic web describing the interactions between the system's constituents. Despite their irreversible and non-equilibrium nature these networks follow Bose statistics and can undergo Bose-Einstein condensation. Addressing the dynamical properties of these non-equilibrium systems within the framework of equilibrium quantum gases predicts that the 'first-mover-advantage', 'fit-get-rich' and 'winner-takes-all' phenomena observed in competitive systems are thermodynamically distinct phases of the underlying evolving networks

Topology of evolving networks: local events and universality

Networks grow and evolve by local events, such as the addition of new nodes and links, or rewiring of links from one node to another. We show that depending on the frequency of these processes two topologically different networks can emerge, the connectivity distribution following either a generalized power-law or an exponential. We propose a continuum theory that predicts these two regimes as well as the scaling function and the exponents, in good agreement with the numerical results. Finally, we use the obtained predictions to fit the connectivity distribution of the network describing the professional links between movie actors.Comment: 13 pages, 3 figure

Giant strongly connected component of directed networks

We describe how to calculate the sizes of all giant connected components of a directed graph, including the {\em strongly} connected one. Just to the class of directed networks, in particular, belongs the World Wide Web. The results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in,out-degree distribution $P(k_i,k_o)$. We show that if $P(k_i,k_o)$ does not factorize, the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in- and out-components. The calculations of the relative sizes of all the giant components are demonstrated using the simplest examples. We explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one.Comment: 4 pages revtex, 4 figure

A Geometrical Test of the Cosmological Energy Contents Using the Lyman-alpha Forest

In this Letter we explore a version of the test of cosmological geometry proposed by Alcock and Paczynski (1979), using observations of the Lyman-alpha forest in the spectra of close quasar pairs. By comparing the correlations in absorption in one quasar spectrum with correlations between the spectra of neighboring quasars one can determine the relation of the redshift distance scale to the angle distance scale at the redshift of the absorbers, $z \sim 2 - 4$. Since this relationship depends on the parameters of the cosmological model, these parameters may be determined using the Lyman-alpha forest. While this test is relatively insensitive to the density parameter $\Omega_m$ in a dust-dominated universe, it is more sensitive to the presence of a matter component with large negative pressure (such as a cosmological constant $\Lambda$) and its equation of state. With only 25 pairs of quasar spectra at angular separations $0.5' - 2'$, one can discriminate between an $\Omega_m = 0.3$ open universe ($\Lambda=0$) and an $\Omega_m = 0.3$ flat ($\Lambda$-dominated) universe at the $4-\sigma$ level. The S/N can be enhanced by considering quasar pairs at smaller angular separations, but requires proper modeling of nonlinear redshift space distortions. Here the correlations and redshift space distortions are modeled using linear theory.Comment: 13 pages, 2 ps figures, submitted to ApJ

Effect of the accelerating growth of communications networks on their structure

Motivated by data on the evolution of the Internet and World Wide Web we consider scenarios of self-organization of the nonlinearly growing networks into free-scale structures. We find that the accelerating growth of the networks establishes their structure. For the growing networks with preferential linking and increasing density of links, two scenarios are possible. In one of them, the value of the exponent $\gamma$ of the connectivity distribution is between 3/2 and 2. In the other, $\gamma>2$ and the distribution is necessarily non-stationary.Comment: 4 pages revtex, 3 figure

Scaling properties of scale-free evolving networks: Continuous approach

Scaling behavior of scale-free evolving networks arising in communications, citations, collaborations, etc. areas is studied. We derive universal scaling relations describing properties of such networks and indicate limits of their validity. We show that main properties of scale-free evolving networks may be described in frames of a simple continuous approach. The simplest models of networks, which growth is determined by a mechanism of preferential linking, are used. We consider different forms of this preference and demonstrate that the range of types of preference linking producing scale-free networks is wide. We obtain also scaling relations for networks with nonlinear, accelerating growth and describe temporal evolution of arising distributions. Size-effects - cut-offs of these distributions - implement restrictions for observation of power-law dependences. The main characteristic of interest is so-called degree distribution, i.e., distribution of a number of connections of nodes. A scaling form of the distribution of links between pairs of individual nodes for the growing network of citations is also studied. We describe effects that produce difference of nodes. ``Aging'' of nodes changes exponents of distributions. Appearence of a single ``strong'' node changes dramatically the degree distribution of a network. If its strength exceeds some threshold value, the strong node captures a finite part of all links of a network. We show that permanent random damage of a growing scale-free network - permanent deleting of some links - change radically values of the scaling exponents. We describe the arising rich phase diagram. Results of other types of permanent damage are described.Comment: 21 pages revtex (twocolumn), 9 figure

Preferencial growth: exact solution of the time dependent distributions

We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability q) or join an already existing cluster with a probability proportional to the size thereof. We calculate exactly the probability \Pm_i(k,t) that the size of the i-th cluster at time t is k. We analyze the asymptotics, the scaling properties of the size distribution and of the mean size as well as the relation of our system to recent network models.Comment: 8 pages, 4 figure

Weighted Evolving Networks

Many biological, ecological and economic systems are best described by weighted networks, as the nodes interact with each other with varying strength. However, most network models studied so far are binary, the link strength being either 0 or 1. In this paper we introduce and investigate the scaling properties of a class of models which assign weights to the links as the network evolves. The combined numerical and analytical approach indicates that asymptotically the total weight distribution converges to the scaling behavior of the connectivity distribution, but this convergence is hampered by strong logarithmic corrections.Comment: 5 pages, 3 figure